On the orders and genera of quadratic forms containing more than three indeterminates

Let us represent by ƒ1 a homogeneous form or quantic of any order containingn indeterminates; by (α(1) ), a square matrix of ordern ; by (α( ), itsi th derived matrix,i. e . the matrix of order ∟n /∟i ∟n-i = I, the constituents of which are the minor determinants of orderi of the matrix (α(1) ) ; and lastly, by ƒi , a form of any order containing I indeterminates, the coefficients of which depend on the coefficients of ƒ1 . When ƒ1 is transformed by (α(1) ), let ƒi ; be transformed by (α(1) ) ; if, after division or multiplication by a power of the modulus of transformation, the metamorphic of ƒi ; depends on the metamorphic of ƒ1 , in the same way in which ƒi depends on ƒ1 , ƒi is said to be a concomitant of thei th species of ƒ1 . Thus: a concomitant of the 1st species is a covariant; a concomitant of the (n — l)th species is a contravariant; ifn = 2 there are only covariants; ifn = 3 there are only covariants and contravariants; but ifn > 3 , there will exist in general concomitants of the intermediate species. There is an obvious difference between covariants and contravariants on the one hand, and the intermediate concomitants on the other. The number of indeterminates in a covariant or contravariant is the same as in its primitive; in an intermediate concomitant, the number of indeterminates is always greater than in its primitive. Again, to every metamorphic of a covariant or contravariant, there corresponds a metamorphic of its primitive ; whereas, in the case of a concomitant of the intermediate orderi , a metamorphic of the primitive will correspond, not to every metamorphic of the concomitant, but only to such metamorphics as result from transformations the matrices of which are the sth derived matrices of matrices of ordern .